Turing and Machines

Abstract: We will discuss newly defined machines that out-perform Turing machines. In his unpublished 1948 paper, Intelligent Machinery, Alan Turing identified several types of machines, with one dichotomy that is false, between active and controlling machines. This mistake became an assumption in his famous 1950 paper and has probably been the source of the confusion about the Turing Test and the Chinese Room Argument. I'll introduce a new type of machine and define a subtype that cannot be simulated by a deterministic Turing machine.

Canceled

Topological dimension through the lens of Baire category

Abstract: In 1913, Brouwer proved that the topological dimension of $\mathbb{R}^n$ is $n$, which implies that there is no continuous injection of $\mathbb{R}^{n+1}$ into $\mathbb{R}^n$. More recently, Izzo and Li wondered if the last statement survives when the requirement of injectivity is relaxed to being injective on a large set. They showed in 2013 that the answer is negative when the largeness is measure-theoretic, i.e. when the continuous function is required to be injective only on a conull set. However, they conjectured that the answer should be positive for the notion of largeness provided by Baire category, namely: there does not exist a continuous function $\mathbb{R}^{n+1} \to \mathbb{R}^n$ that is injective on a comeager set. We will discuss this conjecture and its dramatic (for the speaker) resolution.

A Descriptive Set in Topological Dynamics

Abstract: Let $X$ be a compact metric space and $H(X)$ the group of homeomorphisms of $X$, a Polish group. Then the orbit of $x \in X$ under $h \in H(X)$ (the two-sided orbit) has an obvious definition; its closure is called the orbit-closure of $x$. When all the orbit-closures are minimal then $h$ is called "sharp". (Every orbit-closure contains a minimal one). There are two main theorems. A. The set $S(X)$ of sharp homeomorphisms is always co-analytic. B. For a certain $X$, $S(X)$ is not Borel. In the proof of B we need a variant of Hurewicz' theorem (1930) on the class of uncountable compact sets. This variant should be (but so far isn't) a consequence of Hurewicz' theorem. I'll say a very few words about a new method of proof.

Strong conceptual completeness for &aleph;0-categorical theories

Abstract: Suppose we have some process to attach to every model of a first-order theory some (permutation) representation of its automorphism group, compatible with elementary embeddings. How can we tell if this is "definable", i.e. really just the points in all models of some imaginary sort of our theory? In the '80s, Michael Makkai provided the following answer to this question: a functor $\mathrm{Mod}(T) \to \mathrm{Set}$ is definable if and only if it preserves all ultraproducts and all "formal comparison maps" between them, called ultramorphisms (generalizing e.g. the diagonal embedding into an ultrapower). This is known as strong conceptual completeness. Any general framework which reconstructs theories from their categories of models should be considerably simplified for $\aleph_0$-categorical theories. Indeed, we show: If $T$ is $\aleph_0$-categorical, then $X : \mathrm{Mod}(T) \to \mathrm{Set}$ is definable, i.e. isomorphic to ($M \mapsto \psi(M)$) for some formula $\psi \in T$, if and only if $X$ preserves ultraproducts and diagonal embeddings into ultrapowers. This means that all the preservation requirements for ultramorphisms, which a priori get unboundedly complicated, collapse to just diagonal embeddings when $T$ is $\aleph_0$-categorical. We show this definability criterion fails if we remove the $\aleph_0$-categoricity assumption, by constructing examples of theories and non-definable functors $\mathrm{Mod}(T) \to \mathrm{Set}$ which exhibit this.

Some aspects of simple proofs informed by history, ancient and modern

Abstract: We'll look at (i) the existence of a direct proof (even within intuitionistic logic) of the Steiner-Lehmus theorem, stating that a triangle with two congruent angle bisectors must be isosceles, (ii) the most primitive arithmetic capable of proving the irrationality of $\sqrt{2}$, going back to the early Pythagoreans, and why it cannot prove the irrationality of $\sqrt{17}$, the case Theodorus of Cyrene apparently stumbled upon, and (iii) the simplest proof for the fact that 30 is the greatest number all of whose totitives are prime.

Weak containment in ergodic theory and representation theory

Abstract: The relation of weak containment for unitary representations of locally compact groups is a very useful tool in comparing such representations. Recently Kechris introduced an analogous definition of weak containment for measure-preserving actions of countable discrete groups. We will discuss the relationship between these concepts, and present a result showing that weak containment of measure-preserving actions is an essentially stronger notion than weak containment of the corresponding Koopman representations.

Finite versus infinite: An intricate shift

Abstract: The Borel chromatic number — introduced by Kechris, Solecki, and Todorcevic (1999) — generalizes the chromatic number on finite graphs to definable graphs on topological spaces. While the $G_0$ dichotomy states that there exists a minimal graph with uncountable Borel chromatic number, it turns out that characterizing when a graph has infinite Borel chromatic number is far more intricate. Even in the case of graphs generated by a single function, our understanding is actually very poor. The Shift Graph on the space of infinite subsets of natural numbers is generated by the function that removes the minimum element. It is acyclic but has infinite Borel chromatic number. In 1999, Kechris, Solecki, and Todorcevic asked whether the Shift Graph is minimal among the graphs generated by a single Borel function that have infinite Borel chromatic number. I will explain why the answer is negative using a representation theorem for $\Sigma^1_2$ sets due to Marcone.

Polish groupoids and continuous logic

Abstract: It is well-known that every non-Archimedean Polish group is isomorphic as a topological group to the automorphism group of a countable structure, and analogously, that every Polish group is isomorphic to the automorphism group of a separable metric structure. We will present a generalization of this result: every open locally Polish groupoid admits a full and faithful Borel functor to the groupoid of metric L-structures on the Urysohn sphere, for some countable metric language L. This partially answers a question of Lupini. We will also discuss the analogous result in the non-Archimedean case.

Transcendence bases in ZF

Abstract: It is consistent with ZF+DC that the reals have a Hamel basis but the complex numbers have no transcendence basis. The proof relies on the difference between modularity of the two associated pre-geometries.

Interpolative fusions - preservation results

Abstract: Fix languages $L_1$ and $L_2$ with intersection $L_\cap$ and union $L_\cup$. An $L_\cup$ structure $M$ is interpolative if whenever $X_1$ is an $L_1$-definable set and $X_2$ is an $L_2$-definable set, $X_1$ and $X_2$ intersect in $M$ unless they are separated by an $L_\cap$-definable set. Examples of interpolative structures abound in model theory, but only recently did Minh Tran and Erik Walsberg begin studying the class in the abstract. When $T_1$ is an $L_1$ theory and $T_2$ is an $L_2$ theory, we are interested in the class of interpolative fusions: interpolative structures which are models of the union theory $T_\cup$. Putting aside the nontrivial question of whether this class is elementary (I will assume that it is, axiomatized by a theory $T^{*}$), I will explain how stability-theoretic assumptions on the base $L_\cap$-theory $T_\cap$ lead to preservation results of the form "If $T_1$ and $T_2$ both satisfy property $P$, then $T^{*}$ satisfies property $P$". This is joint work with Minh and Erik.

Colorings of finite subgraphs of the universal k-clique-free graphs

Abstract: It is a central question in the theory of homogeneous relational structures as to which structures have finite big Ramsey degrees. This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorčevi&cacute; in 2005. An infinite structure $S$ is homogeneous if any isomorphism between two finitely generated substructures of $S$ can be extended to an automorphism of $S$. A homogeneous structure $S$ is said to have finite big Ramsey degrees if for each finite substructure $A$ of $S$, there is a number $n$, depending on $A$, such that any coloring of the copies of $A$ in $S$ into finitely many colors can be reduced down to no more than $n$ colors on some substructure $S'$ isomorphic to $S$. This is interesting not only as a Ramsey property for infinite structures, but also because of its implications for topological dynamics. Prior to work of the speaker, finite big Ramsey degrees had been proved for a handful of homogeneous structures: the rationals (Devlin 1979) the Rado graph (Sauer 2006), ultrametric spaces (Nguyen Van Thé 2008), and enriched versions of the rationals and related circular directed graphs (Laflamme, Nguyen Van Thé, and Sauer 2010). According to Nguyen Van Thé , "so far, the lack of tools to represent ultrahomogeneous structures is the major obstacle towards a better understanding of their infinite partition properties." We address this obstacle by providing new tools to represent the universal $k$-clique-free graphs and developing the necesshary Ramsey theory to deduce finite big Ramsey degrees. The methods developed seem robust enough that correct modifications should likely apply to a large class of homogeneous structures omitting some finite substructures.

Superrigidity and measure equivalence

Abstract: Measure equivalence is an equivalence relation on countable groups introduced by Gromov as a measure theoretic counterpart to the goemetric notion of quasi-isometry. In the first part of this talk I will give a brief introduction to measure equivalence. I will then discuss some new joint work with Lewis Bowen in which we show that the class B, of groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli shifts, is invariant under measure equivalence. As a consequence we show that any nonamenable lattice in a product of noncompact locally compact groups must belong to the class B. This also has implications for entropy: we introduce a new kind of entropy called weak Pinsker entropy, and show that equivalence relations generated by free measure preserving actions of groups in the class B completely "remember" the weak Pinsker entropy of the action.